Characterization of Confidence in Multiphase Flow Predictions

Apr 22, 2012 - Confidence in multiphase flow predictions can influence key decisions on the design, location, and operations of pipelines and associat...
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Characterization of Confidence in Multiphase Flow Predictions Selen Cremaschi,*,† Gene E. Kouba,‡ and Hariprasad J. Subramani‡ †

Department of Chemical Engineering, The University of Tulsa, Tulsa, Oklahoma 74104, United States Chevron Energy Technology Company, Houston, Texas 77002, United States

‡

ABSTRACT: Confidence in multiphase flow predictions can influence key decisions on the design, location, and operations of pipelines and associated integrated equipment, e.g., separation systems. These decisions in turn can profoundly affect system reliability, operability, safety, and overall project economics. There may be no application where the lack of confidence in predictions is felt more strongly than for long-distance tiebacks of pipelines to subsea processing in deep water. The uncertainties and resultant confidence in simulator outputs of the flow regime, pressure drop, entrainment, and in situ velocities dictate the estimated confidence levels for opposing and complementary systems and are key contributors in a decision analysis exercise. Uncertainties in benchmarking data, input data to simulator, and internal models for various phenomena propagate to the simulator outputs of the flow regime, pressure drop, entrainment, and in situ velocities. How these discrepancies are treated will affect the scaling of the simulator to different conditions, where benchmarking data may not exist. This paper aims to raise the awareness of the importance and value of uncertainty analysis in multiphase flow systems, provide a methodology with examples, and identify some of the key gaps in the uncertainty analysis.

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INTRODUCTION Subject matter experts are often asked to express a degree of confidence in a prediction, analysis, or design that they or someone else performed. It is sometimes the case that a qualitative degree of confidence can be determined on the basis of years of experience with similar systems. Developing a quantified level of confidence can be difficult because of the lack of uncertainty information on input data, benchmarking data, model predictions, etc. Unfortunately, there may also be an unwillingness to conduct the uncertainty analyses needed to quantify a level of confidence feeling that the effort costs more than it is worth or that it is a hopeless cause because of the aforementioned lack of information. Even so, there are numerous examples of mistakes and overconfidence in designs where reasonable data did exist to assess the uncertainties and level of confidence had the exercise been performed. An accurate quantitative level of confidence is highly desirable for systems that are “too costly to fail” because of the consequences of system failure and/or cost of correcting the system. This is often the situation with remote offshore systems, especially subsea systems in deep water. We claim that a quantifiable confidence in multiphase predictions is needed when designing and determining the operational envelopes for pipelines, separators, pumps, pigging systems, and other equipment dependent upon multiphase calculations. Subject matter experts should be able to provide a reasonable estimate of the range of operational conditions for which there is 90% (or other specified percentage) confidence that the multiphase predictions are valid. Multiphase flow occurs in many systems encountered in chemical, petroleum, and nuclear industries. Because efficient design and operation of these varied systems depend upon the ability to predict the flow characteristics, numerous multiphase flow models and simulator packages have been developed. These models are typically based on a mechanistic approach, which uses the conservation equations of mass, momentum, © 2012 American Chemical Society

and energy along with the empirical or semi-mechanistic closure relationships and adjustable parameters. The resulting equations are solved using various numerical methods and algorithms. The inputs to these models include fluid properties (density, viscosity, and surface tension usually at standard conditions), flow geometry (diameter, wall roughness, and inclination of the pipeline), and operating conditions (gas and liquid flow rates, inlet pressure, and temperature). The usual outputs include flow patterns and profiles of pressure, temperature, in situ velocities, and area fractions of each phase along the pipeline. Multiphase flow models are used extensively to design and operate multiphase flow pipelines with considerable success. However, new applications in the petroleum industry are challenging the acceptable uncertainty of the multiphase predictions especially for extreme conditions. For example, such applications may include remote, long-distance tiebacks of very large pipelines traversing steep terrain in deep water. In many cases, the effects of extreme diameter, pipe length, pressure, and temperature on the uncertainties in the multiphase predictions are not well-established. Furthermore, the results of the multiphase predictions will often influence the design and operation of equipment integrated with the pipeline, e.g., subsea separators, pumps, meters, etc. The net result is that the uncertainties in multiphase predictions have a profound effect on the design and operation of not only the pipeline but also the integrated system. Because of the extreme remote nature of these applications, the integrated system will not be tested at in situ conditions prior to installation, and the costs to live with a flawed design or to correct design errors can be Special Issue: Upstream Engineering and Flow Assurance (UEFA) Received: January 31, 2012 Revised: April 18, 2012 Published: April 22, 2012 4034

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can be smaller than Vc. The blue dashed line in Figure 1 is the pdf for VLf − Vc. Both VLf and Vc are functions of the liquid film height and inversely correlated. In this example, a correlation coefficient of −0.94 between VLf and Vc was used. The fraction of the area under the VLf − Vc pdf for which VLf − Vc < 0 is the probability of sand deposition. The integral of the VLf − Vc pdf is the cumulative probability of deposition, P. Figure 2 shows

extraordinarily expensive. The challenge is to design and operate systems with a quantifiable degree of confidence. There are many sources of uncertainty in multiphase predictions, including the model inputs, assumptions, equations, and the computational methods used to solve the governing equation sets. However, most simulators are not equipped to represent and propagate the source uncertainties and quantify the resulting uncertainty in the model predictions. Therefore, a methodology is required to perform an uncertainty analysis on the existing multiphase flow models. One part of such a methodology is the comparison of simulation results against laboratory and field data. Most often, neither the uncertainties in the predictions nor the comparison data are well-defined. Benchmarking of models against field data may be suspected because uncertainties and biases in the data are typically neglected. Furthermore, there is no industry consensus on a logical methodology to predict uncertainties for scale-up in size, pressure, temperature, fluid properties, or velocities beyond prior experience. Consequently, the industry does not have a methodology to estimate the confidence envelope for predictions in new applications requiring extrapolation beyond current experience. Pipeline design may be governed by the predicted pressure drop, heat transfer, and variety of other transport properties. For instance, erosion because of sand impingement may dictate the upper range of normal flow rates, while sand deposition may set the lower range. The limiting conditions for both erosion and sand transport are typically governed by critical velocities of the carrier fluid. Operating outside of either of these limits can have significant consequences in safety, maintenance, operational life of the pipeline, economics, etc. The uncertainties in these critical velocities are significant in single-phase flow and become very large in multiphase flow. However, it is the difference between the critical velocity and the actual in situ velocity of the carrier fluid that determines the uncertainty in predicting solid transport. Furthermore, the uncertainty of the sand carrier fluid velocity in multiphase flow can dominate the sand transport uncertainty. Figure 1

Figure 2. Cumulative probability of VLf > Vc parametrized on normalized standard deviations of VLf and Vc as functions of VLf/Vc.

the cumulative probability of sand transport (1 − P) as a function of VLf/Vc for a constant Vc and a variety of normalized standard deviations. The common normalized standard deviation is the coefficient of variation, i.e., standard deviation divided by the mean value. The blue dotted line refers to the same conditions as in Figure 1, while the other lines show the effects of increased standard deviations. Several points can be made here: (1) Even with the relatively optimistic uncertainties (low standard deviations) of VLf and Vc, a 90% confidence (P90) in sand transport is not achieved until VLf is ∼50% greater than Vc. (2) Increasing the uncertainties of critical and liquid film velocities to typical values of 50 and 25%, respectively, pushes the P90 film velocity to 2.1Vc, while further increases of velocity uncertainties to 50% requires VLf to be 3.8 times greater than Vc to achieve P90 confidence in sand transport. (3) A very significant observation with respect to future work on reducing sand transport uncertainty is that a reduction in the normalized standard deviation of VLf has a much greater impact on the P90 than the same percentage reduction in the normalized standard deviation of Vc. The implication is that reducing the uncertainty of liquid film velocity predictions in multiphase f low has a greater value for the industry than similar reduction in the uncertainty of critical transport velocity predictions. The costs of uncertainty in deepwater pipelines can be enormous. In addition to the pipe itself, the difference in total installed costs between pipe sizes may include costs associated with different equipment required to install the pipeline, different routing requirements, etc. In deep water, the difference in costs between consecutive pipe sizes can easily run into multiple millions of dollars. Even such a rough understanding of uncertainty in multiphase systems can support the assessment of a reasonable confidence level and provide insight on prioritizing work to further reduce uncertainties and improve confidence in the design of pipelines and associated systems.

Figure 1. Probability density function for key transport velocities, Vc, VLf, and VLf − Vc. σVc/Vc and σVLf/VLf = 25 and 15%, respectively.

illustrates estimated probability density functions (pdf) for the critical transport velocity, Vc, the liquid film velocity, VLf, and the difference between the two velocity distributions. Although the mean of VLf is 25% larger than the mean of Vc, the standard deviations are large enough to cause significant overlap of the distributions, indicating the possibility that VLf 4035

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Figure 3. Uncertainties in pipeline operating points and operational envelope.

Figure 3 represents an operational envelope for a multiphase pipeline. The “+” symbols indicate expected rates over time, and the blue-shaded parallelogram represents the range of possible flow rates that might occur. This region may also combine flow rate uncertainties with expected short-term fluctuations, e.g., slug flow, pipeline sweeps, etc. The solid lines represent limiting conditions, such as sand transport and erosion, for the acceptable range of flow rates. Many other limiting conditions could also be superimposed on the graph to help define the appropriate operational envelope. The best estimates of upper and lower flow rate limits are indicated by the solid lines in Figure 3, and the uncertainties in these limits are indicated by the dashed lines. These lines are labeled with their respective probabilities. The P90 dashed line for the upper limit indicates a 90% confidence that the actual upper limit is above this line, whereas the P90 dashed line for the lower limit indicates a similar confidence that the actual lower limit is below this line. The green region represents the range of possible flow rates for which there is 90% or greater confidence that the flow will not violate either of the limiting conditions. This example operational envelope was based on sand transport and erosion, but the same concept could be applied to different applications with different constraints and limiting conditions. The key point is that we should aim to define the operational envelope where a specified performance can be achieved within a desired confidence, and this requires an understanding of the various uncertainties and how to propagate them to the predictions. Similar to the pipeline operational envelope, we can create and overlay the operational envelopes for various pieces of equipment whose performances depend upon the condition of the flow exiting the pipeline. Typically, the uncertainties in the characteristics of the multiphase flow will dominate the uncertainty in the performance of an integrated device, such as a compact separator. The design of a compact separator system will depend upon predictions of flow patterns, slug size, gas holdup in liquid, liquid entrainment in gas, and bubble/ droplet size distribution, to name a few of the key multiphase flow characteristics. As previously mentioned, most multiphase simulators will use a mechanistic approach in which the special distribution of phases, i.e., flow pattern, is generally first determined to set the

geometry of the problem and to select the appropriate closure relationships dependent upon flow distribution. This determination of flow pattern can be an important source of uncertainty for many of the multiphase flow characteristics. Pereyra et al.1 presented a methodology to quantify confidence levels in the predictions of gas/liquid flow pattern transition boundaries and used the methodology to compare the prediction of flow patterns from the Barnea2 unified model against a variety of experimental data from the literature. The analysis was conducted in the usual non-dimensional variables used in flow pattern mapping and indicated broad transition zones of discrepancy between flow pattern prediction and observation around the predicted boundaries. The overall prediction success rate for the Barnea2 unified model was 75%. Given that Pereyra et al.’s database1 was entirely from welldefined laboratory systems, these results can be used to indicate a maximum level of confidence in flow pattern prediction from the Barnea2 unified model based on distance from predicted transitions. A challenge is to estimate the total uncertainties of predicted flow patterns for field applications because field systems will have additional uncertainties as a result of differences in size, fluids, and operating conditions. Operational points or critical conditions that pass into the transition region between flow patterns should consider the uncertainty in the flow pattern prediction. As an example, Figure 4 is a typical dimensional flow pattern map with black lines following boundaries from the Barnea2 unified model. The gray shading (not to scale) represents a transition or uncertainty region around the boundaries that should be considered for operational points close to the transition. A line of constant critical liquid film velocity is indicated by the red dashed line. The red shading around the dashed line indicates the uncertainty in VLf and illustrates the increase in uncertainty as the dashed line passes through the transition region. The transition between annular and intermittent flow is one that is often encountered in wet gas and high gas−liquid ratio (GLR) flows and can be particularly important when evaluating sand transport, erosion, pipeline fatigue, or separation design. Uncertainty in this transition is compounded in upward inclinations with counter-current liquid flow and the generation of unstable waves, which can produce a range of coherent structures, such as slugs, pseudo-slugs, and large waves, that 4036

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METHODOLOGY

The confidence in multiphase flow predictions can be increased by correctly quantifying the uncertainty in the model predictions. The uncertainty quantification in the model predictions requires a thorough uncertainty analysis, which can be carried out in three steps: (i) identify sources and types of uncertainty in the model, (ii) select appropriate mathematical representations for these uncertainties, (iii) choose and apply an appropriate propagation method to quantify the resulting uncertainty. The first step in uncertainty analysis is to identify the sources and types of uncertainty present in the system under study. One source of uncertainty in multiphase flow model predictions is due to input variable uncertainties. To identify which of these input variables contribute to model prediction uncertainty, the input variables should be categorized into one of two categories: “important” and “sensitive” variables. According to Hamby,4 a small change in the value of the “sensitive” variables will result in significant changes in the output variables; i.e., the “sensitive” variables are highly correlated to the output variables. The “important” variables can be defined as the “sensitive” variables whose values may not be known precisely; hence, the uncertainty in “important” variables will translate to uncertainty in the model outputs.4 Clearly, for increasing confidence in model predictions, the uncertainty of the “important” variables should be propagated to the model outputs. The “sensitive” variables in a model can be identified via parameter sensitivity analysis. Some of the most common and straightforward sensitivity analysis techniques are one-at-a-time sensitivity measures and differential sensitivity analysis. To obtain one-at-a-time sensitivity measures, the modeler varies one parameter at a time, keeping the rest of the variables constant, and calculates the effect of that variable on the output parameters. A sensitivity ranking of the variables can be obtained by varying each variable with the same percentage and measuring its effect on the output variables. Although this approach can be applied easily and the computational cost of this approach is low (equal to the number of variables), it should be kept in mind that the results obtained from this method will be local sensitivities. The differential sensitivity analysis uses the partial differentials of the model with respect to each input variable to determine the sensitivity rankings. Ideally, the higher the partial derivative value, the more sensitive the model with respect to the input variable in consideration. The differential sensitivity analysis, which is again computationally efficient, is best for models that are not highly nonlinear. For highly nonlinear models, the sensitivity rankings are again local and might require complex numerical procedures for calculations. Another class of sensitivity analysis approaches that are mostly straightforward to apply but generally computationally expensive are the approaches that use random sampling methods. These approaches generate statistically significant numbers of input variable values using random sampling techniques, such as Monte Carlo, Latin hypercube, etc. The outputs corresponding to each of these inputs are calculated using the model. Then, the inputs and the corresponding outputs are analyzed using statistical approaches, such as correlations, scatter plots, Pearson’s ranking statistics, regression techniques, etc. The main advantages of these approaches are their ability to investigate the overall input space and to showcase the uncertainty in the outputs because of the interaction of the various uncertainties in the inputs. However, they are computationally expensive, requiring the execution of the model many times. Although not used very commonly, there are also sensitivity analyses that partition the input parameter distributions based on the partition of the outputs and use statistical tests to characterize the generated input distributions to identify the differences. Generally, the idea is to check whether the input distribution partitions were originated from the same population or not. If they were, then the model is not sensitive to that variable or else the model output is sensitive to the changes in that variable. The segmentation-based sensitivity analyses are generally “labor-intensive”, and they are mostly useful if there is a specific question in place regarding the sensitivity of an input variable with respect to the model output range. For additional information on different sensitivity

Figure 4. Gas/liquid flow pattern map. Wide boundaries indicate uncertainty in transition predictions. The red dashed line is a line of constant liquid film velocity, and the red region represents uncertainty in VLf.

propagate along the incline. Furthermore, it is often unclear how the experimental investigators distinguished between these two flow regimes near the transition. For instance, it might be supposed that any large structure bridging the pipe, e.g. a pseudo-slug, would be classified as intermittent. Pereyra et al.’s analysis1 indicated that the critical liquid holdup of 24% Barnea2 used in defining this transition had to be increased to 32% to obtain ∼90% confidence that the observed annular flow would not be predicted as intermittent. In addition to uncertainty bands around transitions, there are often situations where more information in a flow pattern map format would aid in evaluating uncertainties. For instance, additional transitions might include onsets to gas and liquid entrainment and ranges where maximum entrainment should be expected. Mantilla et al.3 developed a mechanistic droplet entrainment model for gas−liquid horizontal pipe flow that predicts the gas rate for onset to droplet entrainment, the fraction of liquid entrained in droplets as a function of operating conditions, and the gas rate at which all liquid outside the viscous sublayer is atomized. Similar capability is not yet available for inclined flow or three-phase flow. It should be expected that the uncertainties in correctly predicting flow patterns will propagate through to the uncertainty in pressure drop predictions. Furthermore, additional uncertainties with respect to scale-up in diameter, length, pressure, etc. need to be taken into consideration. This is especially important for long-distance tieback pipelines transporting multiphase flows in deep water. The predictions of the pressure drop and other multiphase flow characteristics in these lines can strongly influence project economics, and predictions of such characteristics can be very sensitive to the anticipated flow pattern. Up to now, the importance and implications of uncertainty in equipment design and operation of multiphase flow systems were highlighted with some examples. In the next section, a three-step procedure that can be followed to identify, quantify, and propagate uncertainty in deterministic multiphase flow models is outlined. For each step, a suite of tools that are currently available in the literature is briefly reviewed. Several examples of uncertainty quantification in multiphase flow models are presented next, and the paper concludes with gaps and recommendations. 4037

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analysis techniques, the reader is directed to an excellent review of this subject area.4 It should be noted that sensitivity analysis will not differentiate between “sensitive” and “important” variables. Therefore, once the “sensitive” variables are identified, the subject matter experts’ knowledge in the domain will be essential to identify which of the “sensitive” variables are also “important”. However, the sensitivity analysis will provide the sensitivity rankings for the “important” variables. Figure 2 illustrates the probability of sand transport as a function of the critical velocity and liquid film velocity. Here, the model output is the probability of sand transport, and the model inputs are the critical velocity and liquid film velocity. The sensitivity analysis reveals that the probability of sand transport is more sensitive to the liquid film velocity than the critical velocity. Although both liquid film velocity and critical velocity are “important” variables (i.e., they are uncertain variables), the sensitivity analysis reveals that the liquid film velocity is higher ranked than the critical velocity. Therefore, if one aims to increase the confidence on sand transport, one needs to first reduce the uncertainty in model predictions for the liquid film velocity when compared to predictions of critical velocity. Once the “important” variables are identified, the types of uncertainty in these variables and the remaining sources of uncertainty in multiphase models need to be identified. Using sensitivity rankings and the expert opinion about the system and model under study, a decision should be made as to which of these uncertainties will be quantified and propagated to the model outputs during this step. It is also important to differentiate between the error and uncertainty. According to Oberkampf et al.,5 error is “a recognizable inaccuracy in any phase or activity of modeling and simulation that is not due to lack of knowledge”. The errors can be acknowledged, such as mathematical modeling assumptions or numerical solutions of the models, or unacknowledged, which might be results of the computer programming errors. In general, the modeler would have an idea about the magnitude of the acknowledged errors, and hence (in theory), it would be possible to quantify their impact on the outputs. However, the only way to reduce (or preferably eliminate) unacknowledged errors in a model is to perform rigorous and careful model verification studies to ensure that what the modeler intended to do with one operation is what is happening in the computer program. Once errors and uncertainties are differentiated, the next step is to classify the uncertainties. There are two types of uncertainty: aleatory and epistemic uncertainty. Aleatory uncertainty, which is also referred to as stochastic uncertainty or variability, results from the natural variations within the system and, therefore, cannot be reduced.6 In other word, aleatory uncertainty is a result of the natural phenomena. Other than characterizing this kind of uncertainty with appropriate probability distributions, there is no way that the value of an aleatory uncertain variable can be defined with a single number. In contrast, epistemic uncertainty is defined as the outcome of knowledge or information gaps about the system and, hence, can be reduced as our understanding of the system improves and/or more data about the system becomes available.6 For example, slug length in intermittent flow is identified as an “important” variable for pressure drop predictions for two-phase flow. The value of the length of an individual slug is an aleatory uncertainty with its probabilistic nature and its general characterization by positively skewed distributions, such as normal, log-normal, and Ward distributions,7 while the uncertainty in slug length distribution moments, such as mean (μ) and standard deviation (σ), is an example of epistemic uncertainty. Once the sources and types of uncertainty that would be quantified are identified, the next step is selecting an appropriate mathematical representation of the uncertainty. The probability theory has been used for representing uncertainty traditionally.8 However, there are numerous emerging approaches to represent uncertainty, such as evidence theory,8,9 possibility theory,8,10 and interval analysis.8,11 It is well-established that the mathematical language for aleatory uncertainty and variability when sufficient data are available is the probability theory.6,12 Although a champion for epistemic uncertainty representation has yet to be identified, two of the uncertainty representation approaches, interval analysis and evidence theory, are

used more extensively than others in the recent literature. Interval analysis is the appropriate approach that can be used to represent and quantify epistemic uncertainty when only upper and lower bounds of an uncertain variable are available,13 which is the case for most of the experimental data collected in the multiphase flow area. The evidence theory was identified as the most promising of the new uncertainty theories, because it can accommodate both aleatory and epistemic uncertainty.6 Therefore, in this paper, brief overviews of the probability theory, interval analysis, and evidence theory for representing uncertainties will be given. An excellent review and comparison of the uncertainty representation theories for epistemic uncertainty can be found elsewhere.6,12a The probability theory assigns probabilities to represent the amount of likelihood associated with event A, which is a subset of the universal set X. This probability of event A is defined as p(A) and satisfies the following axioms within the power set of universe X [power set of universe X is defined as the set of all subsets of X, and it is represented by W (X)]: (1) The probability of event A that is an element of the power set of universe X should be between 0 and 1 [0 ≤ p(A) ≤ 1 for any A ∈ W (X)]. (2) The probability of the universal set is equal to 1 [p(X) = 1]. (3) The probability of the empty set is equal to 0 [p(⌀) = 0]. (4) For mutually exclusive events A and B, the total probability of the union of A and B (the union of A and B is defined as the set of elements that belong to either set A or set B) is equal to the sum of the probability of A and the probability of B [p(A ∪ B) = p(A) + p(B) if p(A ∩ B) = ⌀].8,12a For example, slug length in intermittent flow is generally represented using the probability theory. For given fluid properties, such as pipe and flow characteristics, all possible slug flow lengths form the power set of the slug length universe for that flow. Then, the probability that the slug length will be lower than some value [which can be defined as event A and its probability of occurrence as p(A)] can be represented by continuous probability distributions and is, in general, characterized with positively skewed distributions, such as normal, log-normal, and Ward distributions.7 Interval analysis is a popular uncertainty model when no information other than crisp upper and lower bounds of the uncertain variable are available.13 The interval that a variable belongs to is represented as Y = [yl, yu] = {y ∈ 9 |yl ≤ y ≤ yu}, where yl and yu are the crisp bounds of the interval. The interval Y does not associate any probabilities or value frequencies between the interval bounds.12a It should be noted that this definition is different from the uniform distribution definition. In the latter case, the value of the uncertain variable is equally likely to be in between the upper and lower bounds of the uniform distribution. However, when the value of a variable is defined to be within an interval, the only information available are the upper and lower bounds of that variable; i.e., the variable itself might follow a distribution or not, but we do not have any information to identify it. Let us continue with the slug length example. Interval analysis is not an appropriate uncertainty model for slug length, because it is shown that the slug length behavior can be represented by positively skewed distributions.7 Furthermore, the exact minimum and maximum slug length for a given operating condition would not be known with certainty. In other words, in the case of slug length, more information is available regarding its behavior. The evidence theory, which is a generalization of the classical probability theory, uses two measures of uncertainty: belief (Bel) and plausibility (Pl).14 These two measures can be viewed as the upper and lower bounds of the probability of an event, and they incorporate the uncertainty in the given information. The belief and plausibility can be expressed using Basic Probability Assignment (BPA), which is a function of m that maps the power set of universe X to [0, 1]; i.e., m = W (X) ⇒ [0, 1]. If A represents an event, which is a subset of universal set X, then m(A) gives “the degree of support of the evidential claim that the true alternative is in the set A but not in any subset of A”.15 Here, m(A) satisfies the following axioms of the evidence theory: (1) m(A) ≥ 0 for any A ∈ W (X). (2) m(⌀) = 0. (3) ∑A ∈ W(X ) m(A) = 1. The belief and plausibility is then calculated using 4038

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∑ m(B) B⊆A

Pl(A) =

∑

m(B) (1)

B ∩ A ≠⌀

From these definitions, it can be seen that belief is a measure of the total evidence that supports event A, where plausibility uses evidence that supports event A totally and partially; hence, Pl(A) ≥ Bel(A). Plausibility and belief are linked to each other with Pl(A) + Bel(A̅ ) = 1, where A̅ represents the opposite of the event A. Plausibility and belief are especially useful when there is little or no data available for the value of the uncertainty and/or there is more than one estimate of the uncertainty and they might contain different information. For example, if several subject matter experts estimate the upper and lower bounds of mixture viscosity of wellbore fluids prior to production and their estimates differ, the evidence theory can be used to combine these estimates to obtain a single uncertainty measure for the viscosity. Let us assume that 10 experts provided the following intervals for possible mixture viscosity of wellbore fluids (μm) in centipoise: [2, 6], [2, 8], [2, 20], [4, 8], [4, 12], [10, 16], [10, 20], [14, 16], [14, 20], and [18, 20]. There are 10 different evidence, which translates into each evidence having a BPA of 1/10 = 0.1, assuming equal weights for each evidence. According to eq 1

∑

Bel(2 ≤ μm ≤ 6) =

Figure 5. Example of uncertainty representation with the evidence theory. and, hence, is the lower bound for the correct probability. Because the plausibility and belief in Figure 5 are a combination of several sets of different evidence, they are discrete. The classical probability theory can be viewed as a special case of the evidence theory, where plausibility and belief are equal for event A. The last step is to choose the uncertainty propagation method that will be used in the analysis. Lee and Chen16 grouped uncertainty propagation approaches into five categories: (1) Simulation-based approaches, such as Monte Carlo simulation, Latin hypercube sampling, or adaptive sampling schemas: These approaches rely on generating an input set that is representative of the input uncertainty, computing the resulting output for each element of the input set, and then using the results to construct the output uncertainty distribution. A graphical representation of the Monte Carlo simulation approach can be seen in Figure 6. When used for uncertainty propagation, the Monte Carlo simulation has the advantage of being non-intrusive. It uses the same deterministic governing equations to calculate the model response for the system under study, and it uses a direct sampling from the defined probability distributions of the input uncertainties. If two or more of the input variables are correlated, this can be defined using the correlation coefficient in the Monte Carlo simulation approach. Then, as the samples for uncertain input variables are generated, the sample generator makes sure that the correlations are not violated by the samples. Although these techniques are very powerful in estimating the output uncertainty, for example, they are sometimes used as the reference to assess the capabilities of the other uncertainty propagation approaches; this robustness comes at a computational expense. (2) Local expansion-based approaches, such as Taylor’s series method or perturbation method: One of the most common approaches practiced in multiphase flow model or experimental data uncertainty propagation is to approximate it using Taylor’s series expansion. Using Taylor series expansion, the combined uncertainty uc(y) of the output of a model y = f(x1, x2, ..., xN) because of uncertain input variables x1 to xN can be estimated as

m(B) = m([2, 6]) = 0.1

B ⊆ [2,6]

∑

Bel(2 ≤ μm ≤ 8) =

m(B) = m([2, 6]) + m([2, 8])

B ⊆ [2,8]

+ m([4, 8]) = 0.3

∑

Bel(2 ≤ μm ≤ 12) =

m(B) = m([2, 6]) + m([2, 8])

B ⊆ [2,12]

+ m([4, 8]) + m([4, 12]) = 0.4

∑

Bel(2 ≤ μm ≤ 16) =

m(B) = m([2, 6]) + m([2, 8])

B ⊆ [2,16]

+ m([4, 8]) + m([4, 12]) + m([10, 16]) + m([14, 16]) = 0.6

∑

Bel(2 ≤ μm ≤ 18) =

m(B) = 1.0

B ⊆ [2,20]

∑

Pl(2 ≤ μm ≤ 4) =

m(B) = m([2, 6]) + m([2, 8])

B ∩ [2,4] ≠⌀

+ m([2, 20]) = 0.3

∑

Pl(2 ≤ μm ≤ 6) =

m(B) = m([2, 6]) + m([2, 8])

B ∩ [2,6] ≠⌀

+ m([2, 20]) + m([4, 8]) + m([4, 12]) = 0.5 Pl(2 ≤ μm ≤ 10) =

∑

m(B) = m([2, 6]) + m([2, 8])

B ∩ [2,10] ≠⌀

N

+ m([2, 20]) + m([4, 8]) + m([4, 12]) + m([10, 16]) + m([10, 20]) = 0.7 Pl(2 ≤ μm ≤ 14) =

∑

uc 2(y) =

m(B) = m([2, 6]) + m([2, 8])

+ m([2, 20]) + m([4, 8]) + m([4, 12]) + m([10, 16]) + m([10, 20]) + m([14, 16]) + m([14, 20]) = 0.9

∑

N

N

⎡ ∂f ⎤⎡ ∂f ⎤ ⎥⎢ ⎥u(xi)u(xj)ρij ⎣ ∂xi ⎦⎢⎣ ∂xj ⎥⎦

∑∑⎢

i=1

i=1 j=1 j≠i

(2) in which the partial derivatives are evaluated at the mean values of the uncertain variables. In eq 2, u(xi) is the uncertainty of the input variable xi and ρij is the correlation between variables xi and xj.8 This uncertainty propagation approach, although generally computationally cheaper to apply, should be used with caution when dealing with large input uncertainties and nonlinear models.16 The partial differentials evaluated at the mean values of the uncertain variables might not be applicable for the whole region of the uncertainty range of the input variables and, hence, underestimate or overestimate the actual output uncertainty. Although if the computational time and resources are limited, local expansion-based approaches would provide an initial

B ∩ [2,14] ≠⌀

Pl(2 ≤ μm ≤ 10) =

⎡ ∂f ⎤2 ⎥ u 2(xi) + ⎣ ∂xi ⎦

∑⎢

m(B) = 1.0

B ∩ [2,20] ≠⌀

The resulting cumulative belief and cumulative plausibility functions are given in Figure 5. As seen from Figure 5 and the above calculations, the plausibility combines all (including partial) evidence from the expert estimates and, hence, gives the upper bound for the correct probability distribution. The belief contains only total evidence 4039

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Figure 6. Uncertainty propagation using Monte Carlo simulation. estimate of the output uncertainty around the mean values of the input variables. (3) Most probable point-based methods, such as the ones based on first-order reliability methods or second-order reliability methods: These methods borrow the ideas developed to determine the failure probabilities or the reliabilities of a system under uncertain conditions. They use first- or second-order Taylor series approximations of the model derivative around the most probable failure point at a given reliability level.17 Then, the reliability level is changed incrementally, and the procedure is repeated to obtain the most probable failure point corresponding to that level. The resulting most probable points are used to construct the output variable distribution.17 These approaches, generally more computationally efficient than simulation-based approaches and more computationally demanding than local expansion-based approaches, provide good estimates of the output probability distributions for models that are not highly nonlinear.17,18 However, they lose their computational advantage against simulation-based approaches as the number of uncertain input variables increases.17 (4) Functional expansion-based approaches, such as polynomial chaos expansion or Neumann expansion: Polynomial chaos expansion is recently receiving considerable attention in uncertainty propagation, especially for models that are computationally expensive even for single runs, such as computational fluid dynamics models.19 In polynomial chaos expansion, the uncertainty of the random input variables is approximated by appropriate polynomial bases and then the uncertainties in the output variables are estimated using these polynomial approximations. A more detailed review of the polynomial chaos expansion applications for uncertainty propagation can be found elsewhere.16,19 (5) Numerical integration-based approaches, such as full factorial numerical integration or dimension reduction methods: Given input uncertainty distributions, these approaches estimate the statistical moments of the resulting output uncertainty using direct numerical integration methods. Then, these statistical moments are used to approximate the output uncertainty distributions using empirical distribution systems.16 Lee and Chen16 performed a comparative study of widely used uncertainty propagation methods in terms of their efficiency, moment estimation, and pdf construction capabilities for the output of blackbox-type models. The methods considered in the study included the full factorial numerical integration (category 5), the first-order reliability (category 3), the univariate dimension reduction (category 5), the polynomial chaos expansion (category 4), and the Monte Carlo simulation (category 1). An approximate uncertainty propagation method, Taylor series expansion, was not considered for analysis because of its limited capability to deal with large input variability and nonlinear models.16,20 It was concluded that univariate dimension reduction and full factorial numerical integration are the most robust and accurate methods if the aim of the uncertainty analysis is to calculate the output moments, whereas all methods, except Monte Carlo simulation and polynomial chaos expansion, are limited in their

ability for pdf construction if the resulting pdf has an irregular shape, such as binomial modes. It is also worth noting that, although computationally the most demanding method, Lee and Chen16 used the Monte Carlo simulation results as the benchmark (or the gold standard) output uncertainty to compare the rest of the approaches.

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EXAMPLE ANALYSIS OF MULTIPHASE FLOW MODELS The application of sensitivity analysis and uncertainty quantification and propagation methods has been very limited in the multiphase flow modeling area. In this section, the few studies (that are available in the literature) will be presented to better understand the multiphase flow model behaviors and uncertainties. One of the initial examples of uncertainty quantification for complex multiphase flow models was proposed by Wilkens and Flach.21 Assuming that input uncertainties are distributed normally, the authors suggested using three model evaluations (at low, mean, and high levels of the input variable) and the corresponding probabilities of these three input points to approximate the mean value of the output variable. Then, the variance of the output variable is calculated using the distance between the estimated mean model output and the model outputs obtained at the low, median, and high levels of the input variables and their corresponding probabilities. The case study quantified the uncertainty in Fanning friction factor predictions around a Reynolds number of 1000 with a standard deviation of 100. The calculated uncertainty range for the Fanning friction factor using the suggested approach yielded 0.016 ± 0.0013. The authors compared their approach to the results of the Taylor series expansion-based uncertainty quantification approach, which yields 0.016 ± 0.0016 for the same case study. The uncertainty quantification approach proposed requires only 3n model evaluations, where n is the number of uncertain input variables. It can be viewed as an approximation to the Taylor series approximation approach and, hence, shares the same limitations as local expansion-based approaches for large input uncertainties and nonlinear models. A sensitivity analysis that investigates the impact of slug length on the pressure drop and liquid holdup predictions of three mechanistic models [Ansari et al. mechanistic model,22 Xiao et al. mechanistic model,23 and Tulsa University Fluid Flow Projects (TUFFP) unified model24] for vertical and horizontal flow were performed by Sarica et al.25 They used perturbation analysis, in which four to eight different slug 4040

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length levels were selected arbitrarily and the corresponding pressure drop and liquid holdup predictions for several operating conditions were calculated. Their analysis showed that, in the developed slug flow, the Ansari et al.22 and Xiao et al.23 mechanistic models were insensitive to the average slug length value. A closer look to the model equations of these mechanistic models revealed that they negate any influence of the slug length when calculating the pressure drop and liquid holdup, thereby eliminating any sensitivity toward the slug length in the developed slug region. The Ansari et al.22 mechanistic model showed a weak sensitivity to slug lengths in the developing slug flow region. The analysis also showed that the TUFFP unified model exhibited significant sensitivity to slug length variations up to 30 pipe diameters. Posluszny et al.,26 whose findings was presented at the 7th North American Conference on Multiphase Flow, focused on quantifying the uncertainty resulting from empirical or semiempirical relationships, i.e., closure relationships, which are essential to solve the closure problem in multiphase flow modeling using a similar approach to the three-step methodology suggested in the previous section. The predictions from these relationships are inherently uncertain variables, because there are experimental errors in the data used to estimate the model parameters and it is statistically not possible to calculate the exact values of the model parameters unless the whole population data are available. As such, both model parameters and the predictions from these closure relationships exhibit epistemic uncertainty. This uncertainty in turn may lead to uncertainty in pressure gradient, liquid holdup, and/or flow pattern predictions of the multiphase fluid models. Hence, they identified the closure relationship predictions as the source of uncertainty that was studied. Consistent with the second of the three-step methodology previously outlined, Posluszny et al.26 selected an appropriate mathematical language for representing the epistemic uncertainty present in closure relationship predictions. One of their goals was to understand the impact of uncertainty in each closure relationship on the uncertainty of the multiphase flow model outputs of interest, i.e., pressure gradient and liquid holdup. Because the probability theory is used extensively to perform this type of analysis,4,27 which is also referred as sensitivity analysis, Posluszny et al.26 selected the probability theory to represent the uncertainties in the closure relationship predictions. As for the uncertainty propagation method, Posluszny et al.26 selected Monte Carlo simulation, citing its widespread and straightforward use for uncertainty propagation analysis with black-box-type models and its robust capability to quantify both output distribution moments and the resulting pdf. To carry out the outlined uncertainty analysis of closure relationship predictions, a wrap-around tool [an uncertainty propagation tool (UPT)] that treats deterministic multiphase flow models as black box models was developed. The tool uses probability distributions to represent the uncertainty and uses Monte Carlo simulations to propagate these to model predictions. The developed UPT, given in Figure 7, includes a closure relationship module, a communication module, and a data analysis module. Given the sample size, the closure model, and the uncertainty distribution, the closure relationship module calculates the mean value and generates a set of random values according to the uncertainty distribution for the flow property under consideration. This information coupled by the fluid properties and operating conditions is converted into the

Figure 7. Uncertainty propagation tool for multiphase flow models using Monte Carlo simulation.26

appropriate model input files and passed to the multiphase flow model by the communication module. Using the UPT, the impact of slug length uncertainty on the pressure drop and liquid holdup predictions was investigated for the TUFFP unified model24 for two-phase vertical flow: oil−gas and water−gas. The sensitivity trends were evaluated using 95% confidence intervals (CIs) around the mean values of the outputs, which were calculated as the difference between the 97.5th and 2.5th percentiles. To compare the relative sensitivity of the pressure gradient and liquid holdup to slug length, normalized percent variability was calculated for each operating condition as follows: % ( −dP /dL)var. = (|( −dP /dL)97.5th percentile − ( −dP /dL)2.5th percentile |) /( −dP /dL)mean × 100

(3)

% (liq. holdup)var. = (|(liq. holdup)97.5th percentile − (liq. holdup)2.5th percentile |) /(liq. holdup)mean × 100

(4)

As an example result, Figure 8 summarizes the observed impact of a ±50% uncertainty in average slug length on the total pressure drop and liquid holdup predictions for oil−gas and water−gas vertical two-phase flows. The fluid properties used as input parameters during this study are summarized in Table 1. The pipe was assumed to be smooth, 0.0508 m in diameter (D), at a 90° inclination angle. The average slug length was assumed be distributed uniformly between 8D as the lower bound and 24D as the upper bound. This corresponds to a ±50% variation from the original value of the average slug length (16D) predicted by the original TUFFP model. In Figure 8, each shaded bar represents a different liquid superficial velocity, which is given as the row headings of the table under each graph. The column headings of the table correspond to the gas superficial velocities. The graphical data are given also in the tabular form below each graph to allow for quantitative analysis by the readers. For vertical oil−gas flow, Figure 8c shows that, at vSG = 0.5 m/s, the pressure drop is not sensitive to slug length 4041

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Figure 8. Impact of slug length uncertainty on the total pressure drop and liquid holdup predictions for two-phase vertical flow: (a) pressure drop sensitivity for water−gas flow, (b) liquid holdup sensitivity for water−gas flow, (c) pressure drop sensitivity for oil−gas flow, and (d) liquid holdup sensitivity for oil−gas flow.26

increases. This behavior was also seen in the water−gas system at higher gas superficial velocities. For higher superficial gas velocities of 4.0 and 5.0 m/s, liquid holdup does not show any dependency to slug length for superficial oil velocities studied. The sensitivity trends of pressure drop (Figure 8c) and liquid holdup (Figure 8d) differ considerably in the oil−gas system. While the pressure drop exhibits substantial uncertainty at high gas and liquid velocities to slug length changes, the liquid holdup is practically constant for the corresponding conditions. At first glance, this seems counterintuitive. However, a closer examination of these points reveals that the high uncertainty in the pressure drop is due to liquid film velocity sensitivity to slug length at these operating conditions. In other words, the film velocity gains dominance in determining the pressure drop sensitivity over liquid holdup at high gas superficial velocities. In a more recent example, Holm et al.28 presented a Monte Carlo simulation-based approach to analyze the impact of uncertainties for oil and gas field development projects. The sixstep methodology developed by Holm et al.28a is quite similar to the methodology described in this paper. It starts with the identification of “key flow assurance requirements”. While some of these requirements may be represented by mathematical relationships, some may not. For example, pressure drop and liquid holdup are examples of “key flow assurance requirements” that are defined by mathematical relationships. “Robust and reliable hydrate prevention during all operational modes”28a is an example of soft “key flow assurance requirement”, which is more general in nature. Although Holm et al.28a suggest that their methodology can be used for both types of the “key flow assurance requirements”, the details and the application of the methodology are presented for quantitative “key flow assurance requirements”. The second

Table 1. Fluid Properties Used for the Example (See Figure 8) oil density (kg/m3) water density (kg/m3) gas density (kg/m3) oil viscosity (kg m−1 s−1) water viscosity (kg m−1 s−1) gas viscosity (kg m−1 s−1) oil−water interfacial tension (N/m) gas−oil interfacial tension (N/m) gas−water interfacial tension (N/m)

859.0 970.0 3.0 0.0275 0.001 1.52 × 10−5 0.01638 0.02914 0.07305

uncertainty. However, this is not the case for higher superficial gas velocities. For example, there is up to a 40% change in the 95% confidence interval of the pressure gradient compared to its mean value at vSG = 8.0 m/s. In comparison to water−gas flow, the pressure gradient of oil−gas flow is more sensitive to slug length changes. The pressure drop sensitivities at vSO = 0.3 and 0.5 m/s are essentially constant for each superficial gas velocity, suggesting that liquid superficial velocity has less impact on pressure drop sensitivity at higher values. The sensitivity of the pressure drop to slug length uncertainties at high gas superficial velocities is due to the sensitivity of the frictional pressure drop to slug length changes, which is correlated with the increased film velocity sensitivities. The ±50% slug length uncertainty causes up to 3.5% variation in the liquid holdup predictions for oil−gas vertical flow for the test points (Figure 8d). At constant vSO, liquid holdup becomes more sensitive to changes in the slug length with increasing vSG up to vSO = 2.0 m/s and then it does not show any sensitivity. At vSG = 0.5, 1.0, and 2.0 m/s, liquid holdup becomes less sensitive to slug length changes as superficial oil velocity 4042

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the output uncertainty but can also be used to understand the underlying behavior of the model under that uncertainty.

step is the identification of key uncertain variables via one-at-atime sensitivity analysis, followed by defining span and probability distributions for the uncertain variables. To propagate the uncertainty, Holm et al.28a uses Monte Carlo simulation. The results of the Monte Carlo simulation is used to obtain the probability distributions of the “key flow assurance requirements”. The paper also provides an overview of the common uncertain variables encountered in flow assurance. The second part of this two series paper28b applied the presented approach to study the flow assurance of the 550 km long two-phase flow line from the Shtokman field to the shore. The analysis considered two sources of uncertainty: the input data and the flow model parameters. The uncertain input parameters were the hydrocarbon liquid fraction, gas and liquid densities, gas and liquid viscosities, gas/liquid surface tension, truck line geometry, hydraulic wall roughness, flow line length, internal diameter, arrival pressure, heat-transfer coefficient, and seawater temperature. The uncertain flow model parameters, which are dependent upon the flow simulator selected (they used OLGA 7 as the flow simulator), were the diameter exponent, inclination term factor, inclination term exponent, roughness effect of droplets, scaling of droplet-wetted wall, smooth, wavy, and gravity turbulences for gas, and low and high turbulences for liquid. For each uncertain variable, the authors provided a minimum, a default, and a maximum value, which were used to construct triangular distributions for uncertainty characterization. Their sensitivity analysis revealed that, of the 23 uncertain variables, the liquid content at the low flow rate was sensitive (defined as the output changes at least 15% when the uncertain variable changes from its minimum to its maximum value) to only 13 of them: (in decreasing impact order) the hydrocarbon liquid fraction, liquid density, wavy turbulence for gas, trunklike geometry, arrival pressure, gas density, liquid gas surface tension, liquid viscosity, seawater temperature, low turbulence for liquid, diameter exponent, heat-transfer coefficient, and gravity turbulence for gas. The inlet pressure at high flow rate was sensitive to mainly hydraulic wall roughness. The authors also concluded that the two outputs considered in this analysis did not change at all with the following variables: smooth turbulence for gas, roughness effect of droplet, and scaling of droplet-wetted wall. For uncertainty propagation studies, Holm et al.28b considered the pressure drop, capacity, liquid inventory, and turndown flexibility as the outputs. They presented graphs of liquid content and required inlet pressure distribution around their means as the flow rate changed from 10 to 50 million standard cubic meters per day (MSm3/d). They concluded that the overall uncertainty for flow rate (±7%) is significantly lower than the overall uncertainty in the liquid content (+126/−47%). Their analysis also showed that the pressure drop at a high flow rate and capacity at a maximum pressure drop only change around 7 and 10%, respectively, because of the different combinations of the uncertain parameters. They concluded that a “typical” pressure drop uncertainty estimate of 10% was too conservative for this system. The last two examples of this section highlight the insight gained through a complete simulation-based uncertainty propagation approach, such as Monte Carlo simulation, for studying the impact of uncertainties in multiphase flow models. The simulation-based approaches not only provide estimates of

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CONCLUSION AND RECOMMENDATIONS The state-of-the-art in multiphase pipeline simulation lacks the characterization and tools to propagate the range of uncertainties needed to establish a reliable confidence interval for predictions in a variety of applications, particularly those requiring significant extrapolation. In this paper, the importance of uncertainty quantification is highlighted for designing and operating multiphase flow systems, especially in the oil and gas industry. A three-step systematic approach is presented to propagate uncertainty for deterministic multiphase flow models, and some of the available tools to accomplish each step of the methodology is briefly reviewed. As discussed in the Introduction, the uncertainties in experimental/field data and model predictions of multiphase flow in pipelines can have a significant impact on the efficient design and safe operation of oil and gas producing and processing facilities. As either a researcher who is fully aware of the advancements on multiphase flows or an adept practitioner of multiphase flow simulators in the industry, one can readily point out the obvious gaps in both laboratory/field data measurement and model predictions of multiphase flows: (1) Measurement errors of fluid properties and flow characteristics, such as the pressure drop, holdup, phase distributions, etc., are not always considered and/or reported in analysis by researchers, particularly for field data. Consequently, model uncertainty accounts for most, if not all, of the discrepancy between observed data and predictions. (2) Technology is not up-todate to measure flow characteristics with reasonable levels of certainty, e.g., ambiguity in visualization of multiphase flows, dispersions, etc. (3) Errors in predictions of one-dimensional (1D) multiphase flow models and the associated closure relations are rarely considered or reported in analysis by researchers. (4) There are very few examples of application of uncertainty propagation approaches to multiphase flow models and, hence, limited data to suggest suitable uncertainty propagation methods for multiphase flow models. (5) Uncertainty bands in flow regime transition lines are not often considered or reported in analysis by researchers. (6) Commercial simulators typically do not incorporate the ability to propagate uncertainties of input data and closure relationships, e.g., flow regime transition, interfacial friction factor, droplet entrainment, gas entrainment, oil and water distribution, and slug size, to the end predictions. (7) Scale-up data are needed to gain confidence in extrapolation of models, particularly for large-diameter pipes operating at very high pressures. (8) In the current commercial multiphase flow simulators, the probability of occurrence and the characterization of dispersions, such as foams, mists, emulsions, and slurries, are not well-characterized and the contributions to the uncertainties of some multiphase predictions are often neglected. To address some of these gaps, we identified the following as short-, intermediate-, and long-term goals. Short term: (1) Encourage vendors of multiphase flow simulators to provide their best estimates of uncertainties in closure relationships, assumptions, numerical shortcuts, etc. (2) Encourage simulator vendors to incorporate propagation of uncertainties from input data and internal models. (3) Identify and rank sources of uncertainties for various applications. (4) 4043

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Develop reasonable expectations for confidence levels for a variety of applications. For example, what is the level of acceptable uncertainty for pressure loss in 10, 50, and 100 mile pipelines. (5) Encourage public research to identify and characterize the various types of errors and uncertainties in their experimental data and model development. Intermediate term: (1) Develop and publish applications of the existing methodologies for characterizing and combining multiphase flow uncertainties, and develop and publish novel and improved methodologies where necessary. (2) Identify the types of scale-up data needed to improve confidence in extrapolating our models, estimate value of data, and seek funding. Long term: (1) Acquire sufficient data at scale to achieve the desired reasonable confidence levels for extrapolation.

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μm = mixture viscosity of wellbore fluids σ = standard deviation σVc = standard deviation of critical transport velocity σVLf = standard deviation of liquid film velocity ρij = correlation between variables xi and xj ⌀ = empty set W (X) = power set of universe X, which is defined as the set of all subsets of X

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AUTHOR INFORMATION

Corresponding Author

*Telephone: (918) 631-3422. Fax: (918) 631-3268. E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The partial financial support provided by the Chevron Energy Technology Company and The University of Tulsa is greatly acknowledged.

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NOMENCLATURE 1D = one dimensional A ⊆ B = A is a subset of B A ∩ B = intersection of set A and set B A ∪ B = union of set A and B BPA = Basic Probability Assignment Bel = belief GLR = gas liquid ratio CI = confidence interval D = diameter dP/dL = pressure gradient % (−dP/dL)var. = percent normalized pressure gradient variability % (liq. holdup)var. = percent normalized liquid holdup variability P = cumulative probability of deposition P90 = 90% confidence p(A) = probability of A pdf = probability density function Pl = plausibility TUFFP = Tulsa University Fluid Flow Projects u(xi) = uncertainty of the input variable xi uc(y) = combined uncertainty of the model output UPT = uncertainty propagation tool Vc = critical transport velocity VLf = liquid film velocity vSG = superficial gas velocity vSO = superficial oil velocity vSW = superficial water velocity X = universal set Y = interval yl = lower bound of the interval yu = upper bound of the interval μ = mean 4044

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